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208 ANALYTICAL MECHANICS
new condition states — in order that a conservative system be in equilibrium its potential energy must have a stationary value.
174. Analytical Criterion of Stability. — The equilibrium of a, body is said to be stable if it is not upset when the body is given a small displacement.
Potential energy is a function of coordinates, therefore we can denote the potential energy of a particle at the point (zi, 3/1, Zi) by the functional relation Ui = Ufa,yi,Zi).
Let us suppose the point fa, yi, z^) to be a position of equilibrium of the particle, and investigate the stability of the equilibrium. If the particle is given a displacement dx, the potential energy in the new position becomes
Expanding J72 by Maclaurin's theorem in powers of dx we obtain
where the subscripts after the parentheses denote that after the indicated differentiations are carried out the coordinates x, y, and z must be replaced by xi, yi, and 21, which are the coordinates of the equilibrium position. But since the particle is in equilibrium at the point
Since 6x, the displacement, is small we can neglect all the-terms of the right-hand member of the last equation except the first. This gives